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When should I use the formula below, and when should I use the chain rule? Or does it not matter? I find using chain rule to be much faster and easier to solve.

$$\lim_{x \to 0} \frac{f(x+h)-f(x)}{h}$$

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That should be $x+h$, not $x-h$. – Hurkyl Sep 24 '13 at 0:55
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You will almost never need to actually use the definition of the derivative to calculate it since we have so many quicker methods (which are all proven from the definition). And no matter what method you're using, you should always use the chain rule if you have a composition of functions, no matter what other rule you may be using.

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You should use the chain rule (or whatever the appropriate rules are for your problem) in practice. The formula $$ f'(x)=\lim_{h\rightarrow0}\frac{f(x+h)-f(x)}{h} $$ is a definition of the derivative. (Note that your expression is similar, but it has a few issues.) The chain rule, along with the power rule, product rule, derivative rule, the derivatives of trigonometric and exponential functions, and other derivative rules and formulas, is proven using this (or another) definition of the derivative, so you can think of them as shortcuts for applying the definition of the derivative to more complicated expressions. It's often quicker and more reliable not to reinvent the wheel and use a shortcut instead.

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You should almost never use the definition of the derivative to actually compute a derivative. In almost all cases, you can use the power rule, chain rule, the product rule, and all of the other rules you have learned to differentiate a function.

You should only need to use the limit definition if you have a strangely-defined function that your can't use the rules for, such as a weird piecewise function.

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Once you're through with textbook exercises to ensure you understand what's going on, most of the time that you invoke that formula is not because you're trying to solve a problem about derivatives. Instead, you invoke the formula because you're trying to solve some other sort of problem (e.g. one involving limits), and this formula lets you turn that problem into a problem about derivatives that is easier to solve.

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The definition of the derivative is just for understanding what you are actually doing, If you want to differentiate a function you should use:

The Power Rule: $$f(x) = ax^n \iff f'(x)=anx^{n-1}$$

The Chain Rule: $$f(x) = h(g(x)) \iff f'(x) = g'(x)*h'(g(x)$$

Product Rule: $$f(x)=h(x)g(x) \iff f'(x)=h'(x)g(x)+g'(x)h(x)$$

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